Problem: What is the greatest common divisor of $1729$ and $1768$?
By the Euclidean algorithm, \begin{align*}
\text{gcd}\,(1729, 1768) &= \text{gcd}\,(1729, 1768 - 1729) \\
&= \text{gcd}\,(1729, 39).
\end{align*}Since the sum of the digits of $1729$ is $19$, which is not divisible by $3$, it suffices to check whether or not $1729$ is divisible by $13$. We can find that it is by long division or noting that $12+1 = \boxed{13}$ divides into $1729 = 12^3 + 1^3$ using the sum of cubes factorization.